Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

General formulation of an analytic, Lipschitz continuous control allocation for thrust-vectored controlled rigid-bodies

Frank Mukwege, Tam Willy Nguyen, Emanuele Garone

TL;DR

This work addresses control allocation for overactuated rigid bodies with vectorized thrusters by developing two complementary methods: a Lipschitz-continuous closed-form solution and a convex optimization formulation that handles practical actuator constraints. The key idea is to use the null-space of the allocation mapping to smooth actuator references and avoid singularities, while enforcing positivity and rate constraints via convex constructs. The main contributions are a systematic null-space smoothing technique that yields Lipschitz continuity in actuator orientations and a convex, constraint-aware optimization framework that preserves near-optimality relative to the Moore–Penrose solution. The methods are demonstrated through simulations on a 3DOF surface vessel and a 6DOF tilting-rotor UAV, highlighting robust singularity avoidance and practical applicability for thrust-vectoring platforms.

Abstract

This study introduces a systematic and scalable method for arbitrary rigid-bodies equipped with vectorized thrusters. Two novel solutions are proposed: a closed-form, Lipschitz continuous mapping that ensures smooth actuator orientation references, and a convex optimization formulation capable of handling practical actuator constraints such as thrust saturation and angular rate limits. Both methods leverage the null-space structure of the allocation mapping to perform singularity avoidance while generating sub-optimal yet practical solutions. The effectiveness and generality of the proposed framework are demonstrated through numerical simulations on a 3DOF marine vessel and a 6DOF aerial quadcopter.

General formulation of an analytic, Lipschitz continuous control allocation for thrust-vectored controlled rigid-bodies

TL;DR

This work addresses control allocation for overactuated rigid bodies with vectorized thrusters by developing two complementary methods: a Lipschitz-continuous closed-form solution and a convex optimization formulation that handles practical actuator constraints. The key idea is to use the null-space of the allocation mapping to smooth actuator references and avoid singularities, while enforcing positivity and rate constraints via convex constructs. The main contributions are a systematic null-space smoothing technique that yields Lipschitz continuity in actuator orientations and a convex, constraint-aware optimization framework that preserves near-optimality relative to the Moore–Penrose solution. The methods are demonstrated through simulations on a 3DOF surface vessel and a 6DOF tilting-rotor UAV, highlighting robust singularity avoidance and practical applicability for thrust-vectoring platforms.

Abstract

This study introduces a systematic and scalable method for arbitrary rigid-bodies equipped with vectorized thrusters. Two novel solutions are proposed: a closed-form, Lipschitz continuous mapping that ensures smooth actuator orientation references, and a convex optimization formulation capable of handling practical actuator constraints such as thrust saturation and angular rate limits. Both methods leverage the null-space structure of the allocation mapping to perform singularity avoidance while generating sub-optimal yet practical solutions. The effectiveness and generality of the proposed framework are demonstrated through numerical simulations on a 3DOF marine vessel and a 6DOF aerial quadcopter.

Paper Structure

This paper contains 12 sections, 38 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Problem Description
  3. Lipschitz continuous solution of the CAP in closed-form
  4. Dynamic optimization for constrained control allocation
  5. Numerical examples
  6. Application of the Lipschitz continuous control allocation
  7. A 3DOF surface vessel with 3 azimuth thrusters
  8. A 6DOF quadcopter UAV with tilting rotors
  9. Application of the dynamic optimization for constrained control allocation
  10. A 3DOF surface vessel with 3 azimuth thrusters
  11. A 6DOF quadcopter UAV with tilting rotors
  12. Conclusion

Figures (15)

  • Figure 1: Example of an actuator frame for an azimuth thruster from kongsberg
  • Figure 2: Surface vessel with 3 azimuth thrusters - Body fixed frame $\boldsymbol{B}$ and the actuator-located $\boldsymbol{B_i}$ frames.
  • Figure 3: Surface vessel with 3 azimuth thrusters - The actuator-fixed frames $\boldsymbol{A_{i}}$ associated to the azimuth angles $\beta_i$ produced by a rotation along the actuator-located $y^{B_i}$ axis. The thrust is produced along the actuator-fixed axis $z^{A_i}$. The rotation produced a force in the $x$ and $y$ body-framed axes.
  • Figure 4: Evolution of the azimuth angles $\beta_i$ in the neighbourhood of $\tau_d = 0$. Dashed curves are azimuth angles with solution \ref{['eq:pseudoInv']}, solid curves are the ones with solution \ref{['eq:BestForm']}. The solid black curve is the value of the thrust added in the nullspace direction to perform the smoothing
  • Figure 5: Representation of a X-shaped quadcopter with tilting rotors
  • ...and 10 more figures